Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 361595, 20 pages
doi:10.1155/2011/361595
Research Article
Continuous g-Frame in Hilbert C∗ -Modules
Mehdi Rashidi Kouchi1 and Akbar Nazari2
1
Department of Mathematics, Science and Research Branch, Islamic Azad University,
Kerman 7635131167, Iran
2
Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of
Kerman, Kerman 7616914111, Iran
Correspondence should be addressed to Mehdi Rashidi Kouchi, rashidi@iauk.ac.ir
Received 18 August 2010; Revised 23 November 2010; Accepted 19 January 2011
Academic Editor: H. B. Thompson
Copyright q 2011 M. Rashidi Kouchi and A. Nazari. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We give a generalization of g-frame in Hilbert C∗ -modules that was introduced by Khosravies then
investigated some properties of it by Xiao and Zeng. This generalization is a natural generalization
of continuous and discrete g-frames and frame in Hilbert space too. We characterize continuous
g-frame g-Riesz in Hilbert C∗ -modules and give some equality and inequality of these frames.
1. Introduction
Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer 1 for
study of nonharmonic Fourier series. They were reintroduced and development in 1986 by
Daubechies et al. 2 and popularized from then on. The theory of frames plays an important
role in signal processing because of their importance to quantization 3, importance to
additive noise 4, as well their numerical stability of reconstruction 4 and greater freedom
to capture signal characteristics 5, 6. See also 7–9. Frames have been used in sampling
theory 10, 11, to oversampled perfect reconstruction filter banks 12, system modelling
13, neural networks 14 and quantum measurements 15. New applications in image
processing 16, robust transmission over the Internet and wireless 17–19, coding and
communication 20, 21 were given. For basic results on frames, see 4, 12, 22, 23.
let H be a Hilbert space, and I a set which is finite or countable. A system {fi }i∈I ⊆ H
is called a frame for H if there exist the constants A, B > 0 such that
2 f, fi 2 ≤ Bf 2
Af ≤
1.1
i∈I
for all f ∈ H. The constants A and B are called frame bounds. If A B we call this frame a
tight frame and if A B 1 it is called a Parseval frame.
2
Abstract and Applied Analysis
In 24 Sun introduced a generalization of frames and showed that this includes more
other cases of generalizations of frame concept and proved that many basic properties can be
derived within this more general context.
Let U and V be two Hilbert spaces, and {Vj : j ∈ J} is a sequence of subspaces of V ,
where J is a subset of Z. LU, Vj is the collection of all bounded linear operators from U into
Vj . We call a sequence {Λj ∈ LU, Vj : j ∈ J} a generalized frame, or simply a g-frame, for U
with respect to {Vj : j ∈ J} if there are two positive constants A and B such that
2 Λj f 2 ≤ Bf 2
Af ≤
i∈J
1.2
for all f ∈ U. The constants A and B are called g-frame bounds. If A B we call this g-frame
a tight g-frame, and if A B 1 it is called a Parseval g-frame.
On the other hand, the concept of frames especially the g-frames was introduced in
Hilbert C∗ -modules, and some of their properties were investigated in 25–27. Frank and
Larson 25 defined the standard frames in Hilbert C∗ -modules in 1998 and got a series of
result for standard frames in finitely or countably generated Hilbert C∗ -modules over unital
C∗ -algebras. As for Hilbert C∗ -module, it is a generalization of Hilbert spaces in that it allows
the inner product to take values in a C∗ -algebra rather than the field of complex numbers.
There are many differences between Hilbert C∗ -modules and Hilbert spaces. For example,
we know that any closed subspace in a Hilbert space has an orthogonal complement, but it is
not true for Hilbert C∗ -module. And we cannot get the analogue of the Riesz representation
theorem of continuous functionals in Hilbert C∗ -modules generally. Thus it is more difficult to
make a discussion of the theory of Hilbert C∗ -modules than that of Hilbert spaces in general.
We refer the readers to papers 28, 29 for more details on Hilbert C∗ -modules. In 27, 30,
the authors made a discussion of some properties of g-frame in Hilbert C∗ -module in some
aspects.
The concept of a generalization of frames to a family indexed by some locally compact
space endowed with a Radon measure was proposed by Kaiser 23 and independently by
Ali at al. 31. These frames are known as continuous frames. Let H be a Hilbert space, and
let M; S; μ be a measure space. A continuous frame in H indexed by M is a family h {hm ∈ H : m ∈ M} such that
a for any f ∈ H, the function f : M → C defined by fm
hm , f is measurable;
b there is a pair of constants 0 < A, B such that, for any f ∈ H,
2
2
Af H ≤ f 2
L μ
2
≤ Bf H .
1.3
If M N and μ is the counting measure, the continuous frame is a frame.
The paper is organized as follows. In Sections 2 and 3 we recall the basic definitions
and some notations about continuous g-frames in Hilbert C∗ -module; we also give some basic
properties of g-frames which we will use in the later sections. In Section 4, we give some
characterization for continuous g-frames in Hilbert C∗ -modules. In Section 5, we extend some
important equalities and inequalities of frame in Hilbert spaces to continuous frames and
continuous g-frames in Hilbert C∗ -modules.
Abstract and Applied Analysis
3
2. Preliminaries
In the following we review some definitions and basic properties of Hilbert C∗ -modules and
g-frames in Hilbert C∗ -module; we first introduce the definition of Hilbert C∗ -modules.
Definition 2.1. Let A be a C∗ -algebra with involution ∗ . An inner product A-module or preHilbert A-module is a complex linear space H which is a left A-module with map ·, · :
H × H → A which satisfies the following properties:
1 αf βg, h αf, h βg, h for all f, g, h ∈ H and α, β ∈ C;
2 af, g af, g for all f, g ∈ H and a ∈ A;
3 f, g g, f∗ for all f, g ∈ H;
4 f, f ≥ 0 for all f ∈ H and f, f 0 if and only if f 0.
For f ∈ H, we define a norm on H by f
H f, f
1/2
A . Let H is complete with this norm,
it is called a Hilbert C∗ -module over A or a Hilbert A-module.
An element a of a C∗ -algebra A is positive if a∗ a and spectrum of a is a subset of
positive real number. We write a ≥ 0 to mean that a is positive. It is easy to see that f, f ≥ 0
for every f ∈ H, hence we define |f| f, f1/2 .
Frank and Larson 25 defined the standard frames in Hilbert C∗ -modules. If H be a
Hilbert C∗ -module, and I a set which is finite or countable. A system {fi }i∈I ⊆ H is called a
frame for H if there exist the constants A, B > 0 such that
A f, f ≤
f, fi fi , f ≤ B f, f
i∈I
2.1
for all f ∈ H. The constants A and B are called frame bounds.
A. Khosravi and B. Khosravi 27 defined g-frame in Hilbert C∗ -module. Let U and
V be two Hilbert C∗ -module, and {Vi : i ∈ I} is a sequence of subspaces of V , where I
is a subset of Z and End∗A U, Vi is the collection of all adjointable A-linear maps from U
into Vi that is, T f, g f, T ∗ g for all f, g ∈ H and T ∈ End∗A U, Vi . We call a sequence
{Λi ∈ End∗A U, Vi : i ∈ I} a generalized frame, or simply a g-frame, for Hilbert C∗ -module U
with respect to {Vi : i ∈ I} if there are two positive constants A and B such that
A f, f ≤
Λi f, Λi f ≤ B f, f
2.2
i∈I
for all f ∈ U. The constants A and B are called g-frame bounds. If A B we call this g-frame
a tight g-frame, and if A B 1 it is called a Parseval g-frame.
Let M; S; μ be a measure space, let U and V be two Hilbert C∗ -modules, {Vm : m ∈
M} is a sequence of subspaces of V , and End∗A U, Vm is the collection of all adjointable Alinear maps from U into Vm .
4
Abstract and Applied Analysis
Definition 2.2. We call a net {Λm ∈ End∗A U, Vm : m ∈ M} a continuous generalized frame, or
simply a continuous g-frame, for Hilbert C∗ -module U with respect to {Vm : m ∈ M} if
a for any f ∈ U, the function f : M → Vm defined by fm
Λm f is measurable;
b there is a pair of constants 0 < A, B such that, for any f ∈ U,
A f, f ≤
Λm f, Λm f dμm ≤ B f, f .
2.3
M
The constants A and B are called continuous g-frame bounds. If A B we call this continuous
g-frame a continuous tight g-frame, and if A B 1 it is called a continuous Parseval
g-frame. If only the right-hand inequality of 2.3 is satisfied, we call {Λm : m ∈ M} the
continuous g-Bessel for U with respect to {Vm : m ∈ M} with Bessel bound B.
If M N and μ is the counting measure, the continuous g-frame for U with respect to
{Vm : m ∈ M} is a g-frame for U with respect to {Vm : m ∈ M}.
Let X be a Banach space, Ω, μ a measure space, and function f : Ω → X a measurable
function. Integral of the Banach-valued function f has defined Bochner and others. Most
properties of this integral are similar to those of the integral of real-valued functions for
example triangle inequality. The reader is referred to 32, 33 for more details. Because every
C∗ -algebra and Hilbert C∗ -module is a Banach space thus we can use this integral and its
properties.
Example 2.3. Let U be a Hilbert C∗ -module on C∗ -algebra A, and let {fm : m ∈ M} be a frame
for U. Let Λm be the functional induced by
Λm f f, fm ,
∀f ∈ U.
2.4
Then {Λm : m ∈ M} is a g-frame for Hilbert C∗ -module U with respect to V Vm A.
Example 2.4. If ψ ∈ L2 R is admissible, that is,
∞ 2
ψ γ
dγ < ∞.
Cψ :
γ −∞
2.5
and, for a, b ∈ R and a /
0
1
x−b
ψa,b x Tb Da ψx ψ
,
a
|a|
∀x ∈ R,
2.6
then {ψa,b }a / 0, b∈R is a continuous frame for L2 R with respect to R \ {0} × R equipped with
the measure da db/a2 and, for all f ∈ L2 R,
fx 1
Cψ
∞
da db
Wψ f a, bψa,b x 2 ,
a
−∞
2.7
Abstract and Applied Analysis
5
where Wψ is the continuous wavelet transform defined by
Wψ f a, b f, ψa,b ∞
1
x−b
fx ψ
dx.
a
|a|
−∞
2.8
For details, see 12, Proposition 11.1.1 and Corollary 11.1.2.
3. Continuous g-Frame Operator and Dual Continuous g-Frame on
Hilbert C∗ -Algebra
Let {Λm : m ∈ M} be a continuous g-frame for U with respect to {Vm : m ∈ M}. Define the
continuous g-frame operator S on U by
Sf M
Λ∗m Λm fdμm.
3.1
Lemma 3.1 see 33. Let Ω, μ be a measure space, X and Y are two Banach spaces, λ : X → Y
be a bounded linear operator and f : Ω → X measurable function; then
λ
Ω
fdμ
Ω
λf dμ.
3.2
Proposition 3.2. The frame operator S is a bounded, positive, selfadjoint, and invertible.
Proof. First we show, S is a selfadjoint operator. By Lemma 3.1 and property 3 of Definition
2.1 for any f, g ∈ U we have
Sf, g M
M
Λ∗m Λm fdμm, g
M
∗
f, Λm Λm g dμm f,
Λ∗m Λm f, g dμm
M
f, Sg .
Λ∗m Λm gdμm
3.3
It is clear that we have
Sf, f Λm f, Λm f dμm.
3.4
M
Now we show that S is a bounded operator
≤ B.
f,
Λ
f
dμm
Λ
S
sup Sf, f sup m
m
f ≤1 M
f ≤1
3.5
6
Abstract and Applied Analysis
Inequality 2.3 and equality 3.4 mean that
A f, f ≤ Sf, f ≤ B f, f
3.6
or in the notation from operator theory AI ≤ S ≤ BI; thus S is a positive operator.
Furthermore, 0 ≤ A−1 S − I ≤ B − A/AI, and consequently A−1 S − I
≤ 1; this shows
that S is invertible.
Proposition 3.3. Let {Λm : m ∈ M} be a continuous g-frame for U with respect to {Vm : m ∈ M}
m m : m ∈ M} defined by Λ
with continuous g-frame operator S with bounds A and B. Then {Λ
−1
Λm S is a continuous g-frame for U with respect to {Vm : m ∈ M} with continuous g-frame operator
S−1 with bounds 1/B and 1/A. That is called continuous canonical dual g-frame of {Λm : m ∈ M}.
m : m ∈ M} that is Sf
Proof. Let S be the continuous g-frame operator associated with {Λ
∗
Λ
m Λm fdμm. Then for f ∈ U,
M
SSf
∗
SΛ
m Λm fdμm M
M
M
Λ∗m Λm S−1 fdμm
SS−1 Λ∗m Λm S−1 fdμm
3.7
−1
SS f f.
Hence S S−1 .
Since {Λm : m ∈ M} is a continuous g-frame for H, then AI ≤ S ≤ BI. On other hand
since I and S are selfadjoint and S−1 commutative with I and S, AIS−1 ≤ SS−1 ≤ BIS−1 , and
hence B−1 I ≤ S−1 ≤ A−1 I.
m S−1 Λm S−1 S Λm . In other words {Λm : m ∈ M} and {Λ
m : m ∈ M}
Remark 3.4. We have Λ
are dual continuous g-frame with respect to each other.
4. Some Characterizations of Continuous g-Frames in
Hilbert C∗ -Module
In this section, we will characterize the equivalencies of continuous g-frame in Hilbert C∗ module from several aspects. As for Theorem 4.2, we show that the continuous g-frame is
equivalent to which the middle of 2.3 is norm bounded. As for Theorems 4.3 and 4.6, the
characterization of g-frame is equivalent to the characterization of bounded operator T .
Lemma 4.1 see 34. Let A be a C∗ -algebra, U and V two Hilbert A-modules, and T ∈
End∗A U, V . Then the following statements are equivalent:
1 T is surjective;
2 T ∗ is bounded below with respect to norm, that is, there is m > 0 such that T ∗ f
≥ m
f
for all f ∈ U;
3 T ∗ is bounded below with respect to the inner product, that is, there is m > 0 such that
T ∗ f, T ∗ f ≥ m f, f.
Theorem 4.2. Let Λm ∈ End∗A U, Vm for any m ∈ M. Then {Λm : m ∈ M} be a continuous
g-frame for U with respect to {Vm : m ∈ M} if and only if there exist constants A, B > 0 such that
Abstract and Applied Analysis
7
for any f ∈ U
2
2 A f ≤
Λm f, Λm f dμm
≤B f .
4.1
M
Proof. Let {Λm : m ∈ M} is a continuous g-frame for U with respect to {Vm : m ∈ M}. Then
inequality 4.1 is an immediate result of C∗ -algebra theory.
If inequality 4.1 holds, then by Proposition 3.2, S1/2 f, S1/2 f Sf, f M Λm f,
√
√
Λm fdμm, hence A
f
≤ S1/2 f
≤ B
f
for any f ∈ U. Now by use of Lemma 4.1,
there are constants C, D > 0 such that
C f, f ≤
Λm f, Λm f dμm ≤ D f, f ,
4.2
M
which implies that {Λm : m ∈ M} is a continuous g-frame for U with respect to {Vm : m ∈
M}.
We define
m∈M
Vm g gm : gm ∈ Vm , M
2
gm dμm < ∞ .
4.3
For any f {fm : m ∈ M} and g {gm : m ∈ M}, if the A-valued inner product
is defined by f, g M fm , gm dμm, the norm is defined by f
f, f
1/2 , then
m∈M Vm is a Hilbert A-module see 28.
Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame for U with respect to {Vm :
m ∈ M}, we define synthesis operator T : m∈M Vm → U by; T g M Λ∗m gm dμm for all
g {gm : m ∈ M} ∈ m∈M Vm . So analysis operator is defined for map F : U →
m∈M Vm
by Ff {Λm : m ∈ M} for any f ∈ U.
Theorem 4.3. A net {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-frame for U with respect to
{Vm : m ∈ M} if and only if synthesis operator T is well defined and surjective.
Proof. Let {Λm : m ∈ M} be a continuous
√ g-frame for U with respect to {Vm : m ∈ M}; then
operator T is well defined and T ≤ B because
2 T g M
2
∗
Λm gm dμm
supf∈U, f
1 sup gm , Λm f dμm
M
Λ∗m gm dμm, f
M
f∈U,
f 1
≤ sup ,
g
f,
Λ
f
dμm
g
dμm
Λ
m
m
m
m
M
f∈U,
f 1
≤ B
gm , gm dμm
.
M
M
4.4
8
Abstract and Applied Analysis
For any f ∈ U, by that S is invertible, there exist g ∈ U such that f Sg ∗
λ
λ
gdμm. Since {Λm : m ∈ M} is a continuous g-frame for
M m m
U with respect to
{Vm : m ∈ M}, so {λm g : m ∈ M} ∈ m∈M Vm and T {λm g}m ∈ M M λ∗m λm gdμm f,
which implies that T is surjective.
Now let T be a well-defined operator. Then for any f ∈ U we have
∗
f,
f,
Λ
f
dμm
Λ
f
dμm
Λ
f,
Λ
m
m
m m
M
M
≤ f M
Λ∗m Λm fdμm
M
Λ∗m Λm fdμm f T Λm f m∈M ≤ f T {Λm f}m∈M 4.5
1/2
.
≤ f T f,
Λ
f
dμm
Λ
m
m
M
It follow that M Λm f, Λm fdμm
≤ f
2 T 2 .
On the other hand, since T is surjective, by Lemma 4.1, T ∗ is bounded below, so T ∗ |RT ∗ is invertible. Then for any f ∈ U, we have T ∗ |RT ∗ −1 T ∗ f f, so f
2 ≤ T ∗ f
2 T ∗ |RT ∗ −1 2 .
It is easy to check that
T ∗ : U −→
Vm ,
T ∗ f {Λm : m ∈ M}
m∈M
∗ 2
2
T f {Λm }
.
f,
Λ
f
dμm
Λ
m
m
m∈M
4.6
M
Hence f
2 ≤ M
Λm f, Λm fdμm
T ∗ |RT ∗ −1 2 .
Corollary 4.4. A net {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Bessel√
net for U with respect
to {Vm : m ∈ M} if and only if synthesis operator T is well defined and T ≤ D.
Definition 4.5. A continuous g-frame {Λm ∈ End∗A U, Vm : m ∈ M} in Hilbert C∗ -module U
with respect to {Vm : m ∈ M} is said to be a continuous g-Riesz basis if it satisfies that
1 λm / 0 for any m ∈ M;
2 if K Λ∗m gm dμm 0, then every summand Λ∗m gm is equal to zero, where {gm }m∈K ∈
m∈K Vm and K is a measurable subset of M.
Theorem 4.6. A net {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Riesz for U with respect to
{Vm : m ∈ M} if and only if synthesis operator T is homeomorphism.
Proof. We firstly suppose that {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Bessel net for
U with
respect to {Vm : m ∈ M}. By Theorem 4.3 andthat it is g-frame, T is surjective. If
T f M Λ∗m fm dμm 0 for some f {fm : m ∈ M} ∈ m∈M Vm , according to the definition
0, so fm 0 for any
of continuous g-Riesz basis we have Λ∗m fm 0 for any m ∈ M, and Λm /
m ∈ M, namely f 0. Hence T is injective.
Abstract and Applied Analysis
9
Now we let the synthesis operator T be homeomorphism. By Theorem 4.3 {Λm ∈
End∗A U, Vm : m ∈ M} is a continuous g-frame for U with respect to {Vm : m ∈ M}. It is
obviouse that Λm / 0 for any m ∈ M. Since T is injective, so if T f M Λ∗m fm dμm 0, then
f {fm : m ∈ M} 0, so Λ∗m fm 0. Therefore {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous
g-Riesz for U with respect to {Vm : m ∈ M}.
Theorem 4.7. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame for U with respect to
{Vm : m ∈ M}, with g-frame bounds A1 , B1 ≥ 0. Let Γm ∈ End∗A U, Vm for any m ∈ M. Then the
following are equivalent:
1 {Γm ∈ End∗A U, Vm : m ∈ M} is a continuous g-frame for U with respect to {Vm : m ∈
M};
2 there exists a constant N > 0, such that for any f ∈ U, one has
Λm − Γm f, Λm − Γm f dμm
M
≤ N min Λm f, Λm f dμm, Γm f, Γm f dμm
.
M
4.7
M
Proof. First we let {Γm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame for U with respect
to {Vm : m ∈ M} with bounds A2 , B2 > 0. Then for any f ∈ U, we have
Λm − Γm f, Λm − Γm f dμm
M
2
Λm − Γm f m∈M 2 Γm f
≤ Λm f
m∈M
m∈M
2
f,
Λ
f
dμm
f,
Γ
f
dμm
Λ
Γ
m
m
m
m
M
M
B2 f 2
≤
f,
Λ
f
dμm
Λ
m
m
4.8
M
B2 ≤
f,
Λ
f
dμm
f,
Λ
f
dμm
Λ
Λ
m
m
m
m
A 1
M
M
B2 .
1
f,
Λ
f
dμm
Λ
m
m
A1
M
Similarly we can obtain
B1 Λm f, Λm f dμm
Λm − Γm f, Λm − Γm f dμm ≤ 1 .
A
2
M
M
Let N min{1 B2 /A1 , 1 B1 /A2 }; then inequality 4.7 holds.
4.9
10
Abstract and Applied Analysis
Next we suppose that inequality 4.7 holds. For any f ∈ U, we have
2 2
{Λm f}
A1 f ≤ f,
Λ
f
dμm
Λ
m
m
m∈M
M
2 2
≤ Λm − Γm f m∈M Γm f m∈M Γm f, Γm f dμm
Λm − Γm f, Λm − Γm f dμm M
M
4.10
≤ N
f,
Γ
f
dμm
f,
Γ
f
dμm
Γ
Γ
m
m
m
m
M
M
.
N 1
f,
Γ
f
dμm
Γ
m
m
M
Also we can obtain
2 2
Γm f, Γm f dμm
≤ Λm f m∈M Λm − Γm f m∈M
M
f,
Λ
f
dμm
−
Γ
−
Γ
dμm
Λ
f,
Λ
f
Λ
m
m
m
m
m
m
M
M
≤ N
Λm f, Λm f dμm Λm f, Λm f dμm
M
M
≤ B1 N 1f 2 .
N 1
f,
Λ
f
dμm
Λ
m
m
M
4.11
Next we will introduce a bounded operator L about two g-Bessel sequences in
Hilbert C∗ -module. The idea is derived from the operator SV W which was considered for
fusion frames by Găvruţa in 35. In this paper, we will use the operator L to characterize
the g-frames of Hilbert C∗ -module further. Let {Λm ∈ End∗A U, Vm : m ∈ M} and
{Γm ∈ End∗A U, Vm : m ∈ M} be two g-Bessel sequences for U with respect to {Vm :
m ∈ M}, with Bessel bounds B1 , B2 > 0, respectively. Then there exists a well-defined
operator
L : U −→ U,
Lf M
Γ∗m Λm fdμm ∀f ∈ U.
4.12
Abstract and Applied Analysis
11
As a matter of fact, for any f ∈ U, we have
M
2
Γ∗m Λm fdμm
sup
M
g∈U,
g 1
2
Γ∗m Λm fdμm, g 2
sup Λm f, Γm g dμm
M
g∈U,
g 1
4.13
≤ sup Λm f, Λm f dμm
Γm g, Γm g dμm
M
g∈U,
g 1
M
≤ B2 Λm f, Λm f dμm
.
M
It is easy to know that L∗ f M
Λ∗m Γm fdμm and L
≤
B1 B2 .
Theorem 4.8. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame for U with respect to
{Vm : m ∈ M}, with g-frame bounds A1 , B1 ≥ 0. Suppose that {Γm ∈ End∗A U, Vm : m ∈ M} is a
continuous g-Bessel net for U with respect to {Vm : m ∈ M}. If L is surjective, then {Γm : m ∈ M}
is a continuous g-frame for U with respect to {Vm : m ∈ M}.
On the contrary, if {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Riesz basis for U with
respect to {Vm : m ∈ M}, then L is surjective.
Proof. Suppose that {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-frame for U with respect
to {Vm : m ∈ M}. By Theorem 4.3, we can define the synthesis operator T of 4.12. It is easy
to check that the adjoint operator of T is analysis operator as follows:
T ∗ : U −→
m∈M
Vm
by T ∗ f {Λm : m ∈ M}
4.14
for any f ∈ U.
On the other hand, since {Γm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Bessel net
for U with respect to {Vm : m ∈ M}, by Corollary 4.4 we also can define the corresponding
operator Q : m∈M Vm → U, Qg M Γ∗m gm dμm.
Hence we have Lf M Γ∗m Λfdμm QT ∗ f for any f ∈ U, namely, L QT ∗ . Since
L is surjective, then for any f ∈ U, there exists g ∈ U such that f Lg QT ∗ g, and T ∗ g ∈
∗
m∈M Vm , it follows that Q is surjective. By Theorem 4.3 we know that {Γm ∈ EndA U, Vm :
m ∈ M} is a continuous g-frame for U with respect to {Vm : m ∈ M}.
On the contrary, suppose that {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous g-Riesz
basis and {Γm ∈ End∗A U, Vm : m ∈ M} is a continuous g-frame for U with respect to {Vm :
m ∈ M}. By Theorem 4.6, T is homeomorphous, so is T ∗ . By Theorem 4.3 Q is surjective,
therefore L QT ∗ is surjective.
12
Abstract and Applied Analysis
Theorem 4.9. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-Riesz basis, {Γm ∈
End∗A U, Vm : m ∈ M} is a continuous g-Bessel net for U with respect to {Vm : m ∈ M}. Then
{Γm : m ∈ M} a continuous g-Riesz basis for U with respect to {Vm : m ∈ M} if and only if L is
invertible.
Proof. We first suppose that L is invertible. Since {Λm ∈ End∗A U, Vm : m ∈ M} is a gRiesz basis, {Γm ∈ End∗A U, Vm : m ∈ M} is a g-Bessel sequence for U with respect to
{Vm : m ∈ M}, by Theorem 4.3 and Corollary 4.4 we can define the operators T, Q mentioned
before and T is homeomorphous, hence T ∗ is also invertible. From the proof of Theorem 4.7
we know that L QT ∗ . Since L is invertible, so is Q. By Theorem 4.6 we have that {Γm ∈
End∗A U, Vm : m ∈ M} is a g-Riesz basis for U with respect to {Vm : m ∈ M}.
Now we let {Λm ∈ End∗A U, Vm : m ∈ M} and {Γm ∈ End∗A U, Vm : m ∈ M} be two
g-Riesz basis for U with respect to {Vm : m ∈ M}. By Theorem 4.6 both T, Q are invertible, so
L QT ∗ is invertible too.
5. Some Equalities for Continuous g-frames in Hilbert C∗ -Modules
Some equalities for frames involving the real parts of some complex numbers have been
established in 36. These equalities generalized in 30 for g-frames in Hilbert C∗ -modules.
In this section, we generalize the equalities to a more general form which generalized before
equalities and we deduce some equalities for g-frames in Hilbert C∗ -modules to alternate
dual g-frame.
In 37, the authors verified a longstanding conjecture of the signal processing
community: a signal can be reconstructed without information about the phase. While
working on efficient algorithms for signal reconstruction, the authors of 38 established the
remarkable Parseval frame equality given below.
Theorem 5.1. If {fj : j ∈ J} is a Parseval frame for Hilbert space H, then for any K ⊂ J and f ∈ H,
one has
2
2
2 2 f, fj − −
f, fj
f, fj fj f, fj fj .
c
c
j∈K
j∈K
j∈K
j∈K
5.1
Theorem 5.1 was generalized to alternate dual frames 36. If {fj : j ∈ J} is a frame,
then frame {gj : j ∈ J} is called alternate dual frame of {fj : j ∈ J} if for any f ∈ H,
f j∈J f, gj fj .
Theorem 5.2. If {fj : j ∈ J} is a frame for Hilbert space H and {gj : j ∈ J} is an alternate dual
frame of {fj : j ∈ J}, then for any K ⊂ J and f ∈ H, one has
2
⎞ ⎝
⎠
−
Re
f, gj f, fj
f, gj fj j∈K
j∈K
⎛
2
⎞ Re⎝
f, gj f, fj ⎠ − f,
g
f
j
j .
j∈Kc
j∈K c
⎛
5.2
Abstract and Applied Analysis
13
Recently, Zhu and Wu in 39 generalized equality 5.2 to a more general form which
does not involve the real parts of the complex numbers.
Theorem 5.3. If {fj : j ∈ J} is a frame for Hilbert space H and {gj : j ∈ J} is an alternate dual
frame of {fj : j ∈ J}, then for any K ⊂ J and f ∈ H, one has
⎛
⎝
f, gj
j∈K
2 ⎛
2
⎞ ⎞ ⎠
⎠
⎝
−
−
f, fj
f, gj fj f, gj f, fj
f, gj fj .
j∈K
j∈Kc
j∈K c
5.3
Now, we extended this equality to continuous g-frames and g-frames in Hilbert C∗ modules and Hilbert spaces. Let H be a Hilbert C∗ -module. If {Λm ∈ End∗A U, Vm : m ∈ M}
is a continuous g-frame for U with respect to {Vm : m ∈ M}, then continuous g-frame {Γm ∈
∗
End∗A U, Vm : m ∈ M} is called
alternate dual continuous g-frame of {Λm ∈ EndA U, Vm :
m ∈ M} if for any f ∈ H, f M f, Γm fΛm fdμm.
Lemma 5.4 see 30. Let H be a Hilbert C∗ -module. If P, Q ∈ End∗A H, H are two bounded
A-linear operators in H and P Q IH , then one has
P − P ∗ P Q∗ − Q∗ Q.
5.4
Now, we present main theorem of this section. In following, some result of this
theorem for the discrete case will be present.
Theorem 5.5. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame, for Hilbert C∗ -module
U with respect to {Vm : m ∈ M} and continuous g-frame {Γm ∈ End∗A U, Vm : m ∈ M} is alternate
dual continuous g-frame of {Λm ∈ End∗A U, Vm : m ∈ M}, then for any measurable subset K ⊂ M
and f ∈ H, one has
K
2
f, Γm ff, Λm f∗ dμm − f, Γm f Λm fdμm
K
2
∗ ∗
f, Γm f f, Λm f dμm − f, Γm f Λm fdμm .
c
j∈Kc
K
5.5
Proof. For
any measurable subset K ⊂ M, let the operator UK be defined for any f ∈ H by
UK f K f, Γm fΛm fdμm.
Then it is easy to prove that the operator UK is well defined and the integral
f,
Γ
m fΛm fdμm it is finite. By definition alternate dual continuous g-frame UK UK c 1.
K
14
Abstract and Applied Analysis
Thus, by Lemma 5.4 we have
f, Γm f
K
2
∗
f, Λm f dμm − f, Γm f Λm fdμm
K
f, Γm f Λm f, f dμm − UK f, UK f
K
∗
∗
∗
UK f, f − UK
UK f, f UK
− UKc UKc f, f
c f, f
5.6
f, UKc f − UKc f, UKc f
2
∗ ∗
f, Γm f f, Λm f dμm − f, Γm f Λm fdμm .
j∈Kc
Kc
Hence the theorem holds. The proof is completed.
Corollary 5.6. Let {Λj ∈ End∗A U, Vj : j ∈ J} be a discrete g-frame, for Hilbert C∗ -module U with
respect to {Vj : j ∈ J}, and discrete g-frame {Γj ∈ End∗A U, Vj : j ∈ J} is alternate dual discrete
g-frame of {Λj ∈ End∗A U, Vj : j ∈ J}, then for any subset K ⊂ J and f ∈ H, one has
f, Γj f
j∈K
f, Λj f
∗
2
−
f, Γj f Λj f j∈K
2
∗
⎝
f, Γj f f, Λj f ⎠ − f, Γj f Λj f .
j∈Kc
j∈K c
⎛
⎞∗
5.7
Corollary 5.7. Let {Λj ∈ LU, Vj : j ∈ J} be a g-frame, for Hilbert space U with respect to {Vj :
j ∈ J} and g-frame {Γj ∈ LU, Vj : j ∈ J} is alternate dual g-frame of {Λj ∈ LU, Vj : j ∈ J}, then
for any measurable subset K ⊂ J and f ∈ H, one has
2
f, Γj f f, Λj f − f, Γj f Λj f j∈K
j∈K
2
⎝
f, Γj f f, Λj f ⎠ − f, Γj f Λj f .
j∈Kc
j∈K c
⎛
⎞
5.8
The following results generalize the results in 30 in the case of continuous g-frames.
Lemma 5.8 see 30. Let H be a Hilbert C∗ -module. If T is a bounded, selfadjoint linear operator
and satisfy T f, f 0, for all f ∈ H, then T 0.
Abstract and Applied Analysis
15
Lemma 5.9 see 30. Let H be a Hilbert C∗ -module. If P, Q ∈ End∗A H, H are two bounded,
selfadjoint A-linear operators in H and P Q IH , then one has
2 3 2 P f, f Qf Qf, f P f ≥ f, f .
4
5.9
Theorem 5.10. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous g-frame, for Hilbert C∗ -module
m : m ∈ M} be the canonical dual continuous g-frame of
U with respect to {Vm : m ∈ M} and let {Λ
{Λm : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one has
Λm f, Λm f dμm K
m SKc f dμm
m SKc f, Λ
Λ
M
Λm f, Λm f dμm Kc
3
≥
4
m SK f dμm
m SK f, Λ
Λ
5.10
M
Λm f, Λm f dμm.
M
Proof. Since S is an invertible, positive operator on U, and SK SKc S, then S−1/2 SK S−1/2 S−1/2 SKc S−1/2 IU . Let P S−1/2 SK S−1/2 , Q S−1/2 SKc S−1/2 . By Lemma 5.9, we obtain
2 2 3 S−1/2 SK S−1/2 f, f S−1/2 SKc S−1/2 f S−1/2 SKc S−1/2 f, f S−1/2 SK S−1/2 f ≥ f, f .
4
5.11
Replacing f by S1/2 f, then one has
3
SK f, f S−1 SKc f, SKc f SKc f, f S−1 SK f, SK f ≥ Sf, f .
4
5.12
On the other hand, we have
SK f, f K
m f dμm m f, Λ
Λ
K
Λ∗m Λm fdμm, f
−1
−1
Λm f, Λm f dμm,
K
Λm S f, Λm S f dμm K
K
K
Λ∗m Λm S−1 fdμm, Λm S−1 f
f, S−1 f
Λ∗m Λm S−1 f, Λm S−1 f dμm
SS−1 f, S−1 f
S−1 f, f .
5.13
Associating with 5.12 the proof is finished.
16
Abstract and Applied Analysis
Corollary 5.11. Let {fm ∈ H : m ∈ M} be a continuous frame for Hilbert C∗ -module H with
canonical dual frame {fm ∈ H : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ H, one
has
f, fm fm , f dμm K
SKc f, fj
fj , SKc f dμm
M
f, fm fm , f dμm Kc
SK f, fj
5.14
fj , SK f dμm.
M
Proof. For any f ∈ U, if we let Λm f f, fj in Theorem 5.10, then we get the conclusion.
Theorem 5.12. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous Parseval g-frame, for Hilbert
C∗ -module U with respect to {Vm : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one
has
K
Λm f, Λm f dμm Kc
2
Λ∗m Λm fdμm
2
3
∗
Λm f, Λm f dμm Λm Λm fdμm ≥ f, f .
4
Kc
K
5.15
Proof. Since {Λm ∈ End∗A U, Vm : m ∈ M} is a continuous Parseval g-frame in Hilbert C∗ module U with respect to {Vm : m ∈ M}, then for any f ∈ U, we have
Λm f, Λm f dμm f, f .
5.16
M
So
Sf, f M
Λ∗m Λm fdμm, f
Λm f, Λm f dμm f, f .
5.17
M
Hence for any f ∈ U, we have S−IU f, f 0. Let T S−IU . Since S is bounded, selfadjoint,
∗
S − IU T , so T is also bounded, selfadjoint. By Lemma 5.8,
then T ∗ S − IU ∗ S∗ − IU
m Λm S−1 Λm . From 5.16, then we have that: for any
we have T 0, namely, S IU , so Λ
Abstract and Applied Analysis
17
measurable subset K ⊂ M and f ∈ U,
Λm SK f, Λm SK f dμm
Λm SK f, Λm SK f dμm M
M
2
SK f, SK f Λ∗m Λm fdμm ,
K
m SKc f dμm m SKc f, Λ
Λ
M
5.18
Λm SKc f, Λm SKc f dμm
M
S f, S f Kc
Kc
Kc
2
Λ∗m Λm fdμm
.
Combining 5.16 and Theorem 5.10, we get the result.
Corollary 5.13. Let {fm ∈ H : m ∈ M} be a continuous Parseval frame for Hilbert C∗ -module H,
then for any measurable subset K ⊂ M and f ∈ H, one has
K
2
f, fm fm , f dμm f, fj fj dμm
Kc
f, fm
Kc
2
fm , f dμm f, fj fj dμm .
5.19
K
Corollary 5.14. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous λ-tight g-frame, for Hilbert
C∗ -module U with respect to {Vm : m ∈ M}, then for any measurable subset K ⊂ M and f ∈ U, one
has
λ
K
Λm f, Λm f dμm 2
λ
Kc
Kc
Λ∗m Λm fdμm
2
Λm f, Λm f dμm Λ∗m Λm fdμm .
5.20
K
Proof. Since {Λm : m ∈ M} is a continuous λ-tight g-frame, then {Λm : m ∈ M} is a
continuous g-Parseval frame, by Theorem 5.12 we know that the conclusion holds.
Corollary 5.15. Let {fm ∈ H : m ∈ M} be a continuous λ-tight frame for Hilbert C∗ -module H
then for any measurable subset K ⊂ M and f ∈ H, one has
λ
K
2
f, fm fm , f dμm f, fj fj dμm
Kc
λ
Kc
f, fm
2
fm , f dμm f, fj fj dμm .
K
5.21
18
Abstract and Applied Analysis
Corollary 5.16. Let {Λm ∈ End∗A U, Vm : m ∈ M} be a continuous λ-tight g-frame, for Hilbert
C∗ -module U with respect to {Vm : m ∈ M}, then for any measurable subset K, L ⊂ M, K L φ
and f ∈ U, one has
! 2 SK L f − SKc \L f 2 SK f 2 − SKc f 2 2λ
Λm f, Λm f dμm.
5.22
L
Proof. Since for any f ∈ U, by Corollary 5.14, we get
! 2 c
SK L f| −SK \L f|2 λ
K
!
λ
Λm f, Λm f dμm − λ
L
Λm f, Λm f dμm λ
K
−λ
K c \L
L
Λm f, Λm f dμm
Λm f, Λm f dμm λ
Kc
2 2
SK f − SKc f 2λ
Λm f, Λm f dμm
Λm f, Λm f dμm
5.23
L
Λm f, Λm f dμm.
L
Acknowledgment
The authors would like to give a special thanks to the referees and the editors for their
valuable comments and suggestions which improved the presentation of the paper.
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